Application of the Gauss-Bonnet theorem to lensing in the NUT metric

Understand

We show with the help of Fermat's principle that every lightlike geodesic in the NUT metric projects to a geodesic of a two-dimensional Riemannian metric which we call the optical metric.

• The optical metric is defined on a (coordinate) cone whose opening angle is determined by the impact parameter of the lightlike geodesic.
• We show that, surprisingly, the optical metrics on cones with different opening angles are locally (but not globally) isometric.
• With the help of the Gauss-Bonnet theorem we demonstrate that the deflection angle of a lightlike geodesic is determined by an area integral over the Gaussian curvature of the optical metric.